DIFFERENTIAL GEOMETRY OF HIGHER ORDER · 2016-12-18 · DIFFERENTIAL GEOMETRY OF HIGHER ORDER 173 A sheaf of %modules is called locally free if it is locally isomorphic to 9 @

Topology Vol. I, pp. 169-Z I 1. Pergamon Press, 1962. Printed in Great Britain

DIFFERENTIAL GEOMETRY OF HIGHER ORDER

WILLIAM FRANCIS POHL~

(Rrceived 30 December 1961)

$1. INTRODUCTION.

$11. FUNDAMENTALS OF HIGHER-ORDER GEOMETRY.

$11.1 Some conventions.

$11.2 Osculating bundles.

$11.3 Osculating spaces.

$111. THE HIGHER-ORDER GEOMETRY OF VECTOR BUNDLES.

§III.l

$111.2

$111.3

$111.4

The differential calculus of vector bundles.

Realization.

Scholium.

Some preliminaries.

$Iv. THE HIGHER-ORDER GEOMETRY OF SUBMANIFOLDS.

3IV.l Submanifolds of Grassmann manifolds and associated maps.

sIV.2 Submanifolds of projective spaces.

gIV.3 Submanifolds of tori.

REFERENCES

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t This is, with some minorrevisions, the author’s thesis in Berkeley; during its preparation he held National Science Foundation Fellowship.

TA 169

170 WILLIAM FRANCIS POHL

$1. INTRODUCTION

IN THIS paper we seek to study geometric constructions which depend on partial derivatives of higher order. This investigation has been suggested in recent years by A. Weil and others [20]; some foundational work has been done by C. Ehresmann [9]. We carry ahead the program and make applications to the study of maps of manifolds, differentiable and complex-analytic.

The first higher-order construction one encounters in geometry is that of the osculating planes to a space curve, a construction which is second-order. A definition of the osculating spaces of order p, for any positive integer p, has been given by E. Cartan [3], the osculating spaces of order one being, of course, the tangent spaces. Now one of the first steps in a development of intrinsic differential geometry is the definition of the tangent bundle of a differentiable manifold M, an abstract bundle which is realized as the concrete bundle of tangent spaces to M when M is realized as a submanifold of an affine space, In like manner, we construct a vector bundle T,(M) over M, which we call the osculating bundle of order p, which is realized as the concrete bundle of osculating spaces of order p when M is realized as a submanifold of an affine space. T, also occurs in the work of Ehresmann and Ambrose et al. We proceed to determine its topological structure.

T,(M) is, naturally, the tangent bundle. To complete the analogy with the first order we prove a generalization to higher order of the Whitney Embedding Theorem.

$111 contains constructions new even for the first order. Suppose V --+ M is a differentiable (resp. complex-analytic) vector bundle over a differentiable (resp. complex-analytic) manifold M and suppose q : V --, kN is a differentiable (resp. complex-analytic) map of V into N-dimensional affine space (with origin), which has the property that its restriction to each fibre is a vector space homomorphism; then we call q a realization of V as a family of planes through the origin in F N. Now suppose V + M is a vector bundle as above; we construct intrinsically, for each positive integer p, a new vector bundle over M, APV, called the p-th derivative of V. The principal geometric property of A,V is the following: let q~ be a realization of V; let A,[cp(V(,)] be the linear span in FN of all the osculating spaces of order p to cp at the points of cp( VI,) (VI, denotes the fibre of V over x E M); then cp induces a canonical realization D,(q) : A,V + FN which has the property that D,(q)(A,VI,) = A,[cp( $)I. The fibre dimension of A,V is n(1 -I- v), where n is the fibre dimension of V, and v is the fibre dimension of T,(M). We find that A,V is topologically associated to V, together with T1, but there is no canonical association; the transition functions of A,V depend on the p-th order partial derivatives of those of V.

The relation of A1 to an analogous construction of Atiyah is discussed in $111.3.

§IV, which might as well be entitled ‘Higher-Oider Enumerative Geometry’, applies the A,, construction to obtain generalizations of the Pliicker formulas and the Todd formula

DIFFERENTIAL GEOMETRY OF HIGHER ORDER 171

of algebraic geometry. Some of the results here are new even for the first order. The main problem is this: suppose M is an algebraic variety in a projective space which has the property that the osculating projective space of order p has maximum dimension everywhere; the p-th associated map, .fo’), is the map (with domain M and range a suitable Grassmann

manifold) which assigns to each point of M the osculating projective space of order p to M at that point; we ask for the homology type off (PI. We solve this problem completely and

thus obtain a generalization to higher order of the Todd formula. When singularities of a certain type occur our method extends and yields generalizations of the Pliicker formulas to higher dimension. The problem is, in fact, formulated and solved for submanifolds of an arbitrary Grassmann manifold. We then study some varieties with degenerate osculating spaces, in particular scrolls, tangent developables, and cones. We conclude the chapter, and the paper, with the formulation of an analogous problem for submanifolds of complex tori and the proof of an analog of the Pliicker formulas for curves in a torus.

It should be noted that the fundamental objects of study here, as in recent work of Chern, Thorn and Grothendieck [6], are not manifolds qua manifolds, but maps of manifolds satisfying geometric conditions, for example a vector bundle or a non-singular map of an algebraic variety into a projective space. In fact, at the present time it seems that the main significance of higher-order constructions lies in their application to the study of maps. It is natural therefore that some notions of classical extrinsic geometry should be taken as

a starting point.

The reader will notice that although $11 and $111 are devoted to different problems, they employ similar methods; we have tried to point out the analogies by similarities of notation. For example, by induction or recursion on the order it becomes scarcely more difficult to prove the propositions for arbitrary order than for second order. The reader will observe the fundamental role played by the symmetric algebra.

The techniques of sheaf theory and homological algebra are heavily exploited in $II and $111. By this means analytic computations are, for the most part, eliminated from the proofs. The prerequisites are to be found in [ 131 and [ 161, except for some special techniques employed in $111.3, which are to be found in [2].

The considerations of this paper lead to many new problems. Some of those which are at the moment susceptible of precise statement are indicated in the course of the work.

The author wishes to thank his teachers. He especially wishes to thank S. S. Ghern, under whose direction this paper was prepared; this work is essentially a response to a

number of questions posed by him. The author also wishes to thank J. Simons for some important suggestions.

PII. FUNDAMENTALS OF HIGHER-ORDER GEOMETRY

gII.1. Some Conventions.

The principal considerations of this paper apply equally to the category of real differentiable manifolds and that of complex-analytic manifolds. In order to have a unified exposi- tion we have chosen to employ a neutral terminology, e.g., ‘manifold’, ‘map’, ‘vector bundle’,

172 WiLLlAM FRANCIS POHL

‘dimension’. If one wants to interpret the statements in the real differentiable case he will read for these terms ‘P-differentiable manifold’, ‘differentiable map’, ‘differentiable real vector bundle’, ‘real dimension’, etc. ; if he wants to interpret the statements in the complex- analytic case he will read for these terms ‘complex-analytic manifold’, ‘holomorphic map’, ‘complex-analytic vector bundle’, ‘ complex dimension’, etc. Those parts which depend on something proper to either case, for example those involving the Whitney Embedding Theorem or the Chern classes, will employ the standard and explicit terminology.

Everything is so formulated that it may be transferred easily to the category of abstract algebraic varieties, provided that in case the base field is of finite characteristic, the latter

be greater than the order.

Techniques of vector bundles will be heavily exploited throughout this paper. Let f” : M + N be a map of manifolds and let V 2 M and W 2 N be vector bundles over M and N respectively. Then we say thatf : V + W is a homomorphism of vector bundles over

,f” iff is a map of manifolds, if the diagram

f V-W

M-N

is commutative and if for each x E M,f : VI, + Wl/‘.Cxj is a vector space homomorphism, where IX denotes the fibre of V over x E M. (This notation for the fibre over x will be used throughout this paper.) We say that x E M is a singular point off if f : Vlx + W is not injective. Iff” : M --) A4 is the identity map, we call a homomorphism of vector bundles overf* simply a homomorphism of vector bundles.

Iff” : M + N is a map of topological spaces and if W -+ N is a vector bundle over N, one constructs, in a standard fashion, a new vector bundle over M, f”‘W + M, called the inverse image of W, and a homomorphismf” : f”! W + W over f “. Iff : V + W is a homomorphism of vector bundles overf”, then there is induced a homomorphism f’ : V + f”! W andf’f’ = 5

In the present and following chapters we shall make extensive use of the theory of sheaves of modules over sheaves of rings, given, for example, in [16]. If M is a manifold we consider, in particular, two sheaves of rings over M, g(M) and La(M). In the real case g(M) is the sheaf of germs of constant real-valued functions on M and 9(k) is the sheaf of germs of P-differentiable functions on M; in the complex case V?(M) is the sheaf of germs of constant complex-valued functions and 52(M) is the sheaf of germs of holomorphic functions. Now S’(M) c 9(M), and hence a sheaf of .9?( M)-modules may also be considered as a sheaf of S’(M)-modules.

We shall frequently take tensor products of &sheaves. These products are to be understood as being over $3 unless the contrary is indicated.


A sheaf of %modules is called locally free if it is locally isomorphic to 9 @ . . . $ 9,

where @ ‘denotes the direct sum of sheaves. A locally free sheaf of .%modules is always the sheaf of germs of sections of a vector bundle, and conversely.

In the complex-analytic case, the term ‘coherent’ is to be interpreted in the usual sense. In the real differentiable case it is to be interpreted as meaning precisely locally free. It is well known in the complex case, and easy to prove in the real case, that an extension of a coherent sheaf by a coherent sheaf is coherent.

gII.2. Osculating Bundles.

Let M be a manifold, %7(M) and g(M) the sheaves just defined, F,(M) = F1 the sheaf of germs of tangent vector fields (differentiable or holomorphic depending on the case under consideration), and T,(M) = T, the tangent bundle of M.

DEFINITION. The sheaf of germs of osculating vector fields of order p of M, F-,(M) = .F,, is the subsheaf of .%‘~IIQ(~, 9) consisting of those germs of %-linear endomorphisms of 9

which can be written as sums of r-fold homomorphism compositions of elements of .4r,, r < p,

i.e. as

Thus, in terms of local co-ordinates xi, . . . , x, on M, .FP consists of all germs of fields

of operators of the form

where the a’s are functions (differentiable or holomorphic).

We shall show that I, is the sheaf of germs of sections of a vector bundle T,(M) = Tr;

Tr is the same as the dual of the bundle ofp-jets of functions in the sense of [9; see also 141. It is also referred to as the bundle of p-th order tangent vectors [ 11. There are various fancy definitions of T,, which generalize Chevalley’s definition of tangent vectors [IO]; however we shall omit them since their geometric significance seems to be slight. We show that if M is realized as a submanifold of an affine space, T,(M) is realized as the concrete bundle of osculating spaces of order p to M, just as T,(M), the abstract tangent bundle of M, is realized as the concrete tangent bundle of tangent spaces to M.

We recall the following [5]:

DEFINITION. Let V be a vector space. For any permutation, u, of p letters, let a : BP V

+ @J’V be the linear endomorphism which sends v1 @ vz @ . . . @ up to z!,,(~) Q vac2) @ .~.

63 t’,(p)* Let S, = l/p! 1 u be the symmetrization operator, where the sum is over all per- mutations o. Then @‘/&er(S,) is called the p-foM symmetric product of V, OpV. S, gives


rise to the canonical projection S,: @V+ 0 p V and to a map S, : 0 p V --f @JR V; we shall denote these three maps by the same letter. The image in 0 pV of u, 0 LJ~ @ . . . @I zjp will be denoted by ui 0 v2 0 . . . 0 up.

Iff : @ pV + W is a symmetric linear transformation with values in a vector space lzl, there exists a natural liftingf : 0°F’ + W, such that the following diagram is commutative :

OT- \ \ \ SP .r\

fl @“V------+ Iv.

Note that all these constructions on V are natural, i.e., they commute with the action of the general linear group, so that they give rise to constructions on vector bundles and sheaves.

THEOREM 2.1. Y-,(M) is the sheaf of germs of sections of a rector bundle T,(M) = Tp, called the bundle of osculating vectors of order p to M, andfor each p > I there is a natural exact sequence of rector bundles

IP PP

2,(M): o+ T,_, + Tp-+OPT,-+O,

hshere I, is the inclusion map.

Proof. We define a g-homomorphism (u is the sheaf of germs of constant functions) m : @p%Fl + ~p/~p-t by

m(DlQ4Q . . . 0 Dp) = P;(D,Dz 1.. DI;>,

where P; is the canonical projection Fp -+ Fp/Yp-l. (When we write @ or 0 without indicating over which sheaf of rings the products are taken, it is to be understood that the sheaf is 52.) Now m is a symmetric map; to show this it suffices to show that ma = m, where cr is a permutation which interchanges two adjacent letters and leaves the others fixed, for any permutation may be written as a product of such. If u interchanges the 6th and i + 1-st letters, then

(m - ma)(D, 0 . . . 0 Dp)

= P;(D, . . . D, - D, . . . Di_1Di+,DiDi+z ... Dp)

= P;(D, . . . Di-,CDiv Di+IlDi+z ..* Dp),

since the Lie bracket of two vector fields is again a vector field.


And m induces a 3homomorphism m : OpTI + Tp/Tp_,; for iff E 9,

I?@), @ . . . QfDiQ **. ODp)

= m(fDi Q D, Q . . . Q Di_1 Q Di+l 0 *a* ODp>

=fm(DiQDIQ . . . QDi-lQDi+iQ -0. QDp)

= fm(D1 Q . . . Q Dp),

where the first and last equalities follow from the already-proved symmetry of m. Thus we have, by the lifting property of symmetric maps, a natural Q-homomorphism A and a commutative diagram of .%homomorphisms of g-sheaves

It is immediate from our definitions that m is onto; hence fi is onto. We must show that n^l is injective.

LetxEMandletx,,..., x, be local co-ordinates on M such that Xi(x) = 0; the elements

-$-I o...O$/, iI<... Gi,, ‘I x ‘P x

form a basis, under the action of 9, of OP$II,. If ker(fi) # 0 at x, then there is some relation of the form

(2.1) xp-llx = 1 ai, *** #pax, aP it b . . . Sip ,, . ..aXi. I x’

where XP _ 1 Ix is an operator of lower order at x and the u’s are not all zero. But if a,, .,. iP # 0, apply both sides of (2.1) to XilXiz . . . Xip, and get 0 = MUi, . . . I,,, for some integer M # 0, which is a contradiction. Since x is an arbitrary point, 61 is injective. Thus the sequence of C&homomorphisms

1, p, o-+Y-P_,~Y--,+o’~l+o

is exact, where Pp = ~ii-‘Pi.

Now F1 is locally free; by induction assume Fp_ 1 is locally free; then since OPTI is locally free and an extension of a locally free sheaf by a locally free sheaf is locally free, .FP is locally free, so it is the sheaf of germs of sections of a vector bundle T,(M) = Tp, q.e.d.


COROLLARY. T,(M) is a rector bundle ofjbre-dimension

v(m, p) = m + (“:‘j+ ,.. +(m+;-11,

where m is the dimension of M.

A word of caution is in order in case A4 is a complex manifold. The tangent bundle of M as a real manifold and the tangent bundle of M as a complex manifold coincide. However

for p > 1 the osculating bundle of order p of M as a real manifold and the osculating bundle of order p of M as a complex manifold do not coincide, for if M has complex dimension m, the real fibre-dimension of the former is 2m t- (2m(2m + 1))/2 while the real fibre-dimension of the latter is 2(m + (m(m + 1))/2).

It is worthwhile to look at the transition functions of T,(M). Let (U, I/, . . ) be a covering of M by co-ordinate neighborhoods. Then with respect to {U, V, . . .) the transition functions of T,(M) from U to V take the form

where Juiy are the transition functions of O’T,(M), A and C are matrices of second partial derivatives of the co-ordinate transformation functions, and B is a matrix of their third partial derivatives.

Let us now examine affine iv-space, F N. We have already established the exact sequence

IP PP Z,(FN) : 0 -+ Tp_,(FN) + Tp(FN) + OpTI + 0.

Letx,, . . . , xN be the co-ordinates of FN; define a vector-bundle homomorphism oP : Tp(FN)

--) Tp - l(FN) by

( xp-l + C ai* . ..i.

d’ Op

JXj, . . . aXi, 1 = X,-l,

where Xp _ 1 E Tp- ,(FN). Clearly op is a splitting of X,(FN). Define

Q7, = oPop-l . . . o2 : Tp(FN) -+ Tl(FN)

and m, = id. Note that the o,‘s, and hence the UJ~‘s, commute with afline transformations of FN.

If M is an arbitrary real differentiable manifold, the exact sequence Z,(M) always splits differentiably; in the case of Z2 the splittings, or ‘disections’, are in natural one-to-one correspondence with the family of connections on M without torsion, according to [l]. But no splitting is prescribed by the differentiable structure of M alone. Thus T,(M) is associated to T,(M), but there is no canonical association. In the complex-analytic case there is in general no analytic splitting. In fact we shall show that in order for an analytic splitting to exist for compact K%hler A4, it is necessary that the Chern classes of M vanish.


Let f : M -+ N be a map of manifolds. Then f induces a map f* : 9(N) --+ 9(M) defined by composing germs of functions on M with J This map is a homomorphism in the sense that if f~ and b belong to the same stalk of 9?(N) and i and p are constants,

*(Aa -i- pb) = if*(a) + pf*(b).

Let f, : &,IIs~(S(M), F) + %%&9(N), F) denote the adjoint map. Now T,(M) is contained in &r&SB(M), F) for any manifold M, for TP arises from F,, by restricting fields of operators; also ~~(~~(~)) c T,(M) and f,lT,(M) is a vector bundle homonlorphism, as one sees from the chain rule. The restriction off,,f, : 7’,,(M) --P T,,(N), we call the p-tk diflerential of $

The differential enjoys the functorial properties: cfg), = fpg,, and (identity)~ = identity.

THEOREM 2.2. 2,(M) is a functor; i.e. iff : M + N is a map of manifolds,

is a commutative diagram. Consequently, iffi is non-singular, so is fp. for &,r p.

Proof. The left-hand rectangle of diagram (2.2) is commutative because f,_l is, in reality, just the restriction off,. To prove the commutativity of the right-hand square we pass to local co-ordinates. Let x E M, let y = f(x), and let x1, . . . , x,,, (resp. yl, . . . , 4;1) be co-ordinates on M (resp. N) valid in a neighborhood of x (resp. y). Then f is given by yi = vi(x), 1 < i Q n. If 40 is a germ of a function on A4 at j

+ lower order derivatives of cp.

Hence

$11.3. Osculating Spaces.

= Ppfp 3

1 aXi, ,.I dXi~, ’ q.e.d.

The following definitions were given by E. Cartan in [3].

DEFINITION. Let U be an open subset of one-dimensio~af a@tte space and let f : U -+ F’ be a curce in N-dimensional afine space. The osculating space of orders p to f at x E U is the plane in F” spanned by the points

f(x), ffx) af’(s),f(-x) -!“f”(.Y), *.. 1 f(x) + f’,p’(x),

where the sum is rector addition.


DEFINITION. Let f : M + FN be a map of a manifold M into N-dimensional ajine space. The osculating space of order p to fat x E M is the plane in FN spanned by all the osculating spaces of order p at x to curves on M through x.

Let mP : T,(FN) --t Tl(FN) be the map defined in the last section and let p be the map which identifies tangent vectors to F’ with points of FN, so that if xi, . . . , xN are the co- ordinate functions of FN, and if a = (a,, . . . , aN) E FN, and if

DE 2b.z i=r ‘aXi (I

is a tangent vector to FN at a, then p(D) = (b, + al, b2 + a,, . . . , bN + aN>. That Tp is the sought-for abstraction of the osculating spaces of order p is contained in

PROPOSITION 2.3. Let M be a manifold and let f : M + FN be a map into N-dimensional space. Then pmpfp : T;(M) + FN maps T,(M)I, onto the osculating space of order p to f at x, for each x E M.

Proof. We prove the proposition first for a curve f(x). If r < p,

= f(x) + j”‘(X). Since T,(M) is spanned, for M a curve, by the d’/dx* for 1 < r < p, the proposition is established in that case.

Now let Mm be an m-dimensional manifold. It follows from the functorial properties of the differential that pzvpfp(Tp(M)lX) contains the osculating space of order p at x. To

prove the proposition in general we require only the following

LEMMA. T&M”) lx is spanned by {g&T,(U)I,.)) for a/l curces g : U + M” such that g(x’) = x.

Proqf. The lemma is clear for p = 1; assume it for p = r - 1. Then if g is a curve as

above, t a local parameter on U about x’, and yi, . . . , y, local co-ordinates on M,

where X,_ 1 is a field of order r - 1 and the Ci,...i, are the integers defined by the formal

expression

where the qI are independent indeterminates. Now TJM”‘) is spanned by the

a’/dxi, . . . cxi,; we have the lemma if we can show that the coefficients of these operators

in (2.4) satisfy no relation for all curves g. But the C’s are non-zero and the dx,g/dt satisfy no relation, for if (a,, . . . , a,,,) is an arbitrary m-tuple, define a curve g by xi(g(t)) = a,t; then

dx,g dt ’ ...’

i, . . . , a,).


Hence the coefficients of the g/ax,, . . . ax,_ in (2.4) satisfy no relation for all curves, and

thereby is the lemma, and thus the proposition proved.

DEFINITION. Let f : Mm + FN be a map of a manifold into an affine space. We say that f is singular qf order p at x E Mm if the homomorphism of vector bundles over f, w,& : T’(M”) + T,(FN), is singular at X.

By Proposition 2.3, fis singular of order p at x if and only if the osculating space of order p to f at .X has dimension less than v(m, p), where v(m, p) is the number defined in the Corollary to Theorem 2.1. Further explanation is in order. Let f : M -+ N be a map of manifolds. Then by Theorem 2.2, iffr : T,(M) + T,(N) is non-singular, i.e., iffis non- singular in the usual sense, so is fp : T,(M) + T,(N). Consequently for a map into affine space, a p-th order singularity is not a singularity of the p-th differential; we must bring the splitting of T,(F’) provided by the affine structure into play.

A second-order singularity of a curve in affine space is called an in$ection point. Let x1 = a,t, x2 = a2t, be a line in the plane; it is non-singular in the first order, but everywhere singular in higher order. Let x1 = t, x2 = t3 be a cubic plane curve. It is everywhere non-singular in first order, but has an inflection point at t = 0. Roughly speaking, then, a higher-order singularity of a submanifold of an affine space is a point where the submanifold fails to curve sufficiently.

Caution. The terminology ‘singularity of order p’ is used in recent literature to denote a degenerate critical point. For p > 1 our singularities of order p are not necessarily topological singularities, as the examples above demonstrate; they are singularities in the sense of ‘special points’; however, we have found a means of representing them as singularities of a vector bundle homomorphism, ro,,fb. But since ml = id., our first-order singularities are precisely the singularities in the usual sense.

THEOREM 2.4. (The Whitney Embedding Theorem for higher order.) Let M be a mangold bvhich ddmits an immersion (resp. embedding) in an afine space. Then for each p, M admits some immersion (resp. embedding) in an afine space which is non-singular of order p.

ProoJ We prove the theorem first for M = FN, N-dimensional affine space. List all unordered r-tuples of integers P = [iI, . . . , i,], r < p, 0 < ij < N, as PI, . . . , PL, (L being equal to v(N, p)) in such a way that if r < s, r-tuples come before s-tuples. For each k define P,,:F’ + F’ by Pk(xIr . . . , xN) = XirXi, . . . xi, where [iI, . . . i,] is the k-th r-tuple. Then the map P : FN -+ FL, the Cartesian product of the Pk’s, is an embedding, since the first N of the PA’s are just the co-ordinate functions of FN.

We assert that P is non-singular of order p; for let x E FN be an arbitrary point and let Dk = ?/?-Xi, . . . a~,~l,, where [iI, . . . , i,] is the k-th r-tuple; then pw,P,(D,) has no component in the t-th direction if t < k, but it has a component in the k-th direction; hence pw,P,(T,(FN)I,) is all of FL. Since L. = v(N, p) = dim(T,(FN)I,), w,P, is non-singular.

Now let M be a manifold which admits an immersion (resp. embedding) f : M + FN. Then Pf : M + FL is non-singular of order p9 for w,(Pf), = w,P,f,, by the functorial property of the p-th differential; but wpPp is non-singular and so is fp, by Theorem 2.2; hence w,P,f, is non-singular, q,e.d.


One can apply the projection argument of the Whitney Embedding Theorem [see, e.g.. 81 to sharpen Theorem 2.4 in the real case. The result is the following: suppose M is a real differentiable manifold; then A4 admits an embedding in (m f v(m, p))-dimensional real affine space which is non-singular of order p, provided that p > 1.

Note that the construction of Theorem 2.4 is analytic, so we also get a higher-order analog of the Morrey Embedding Theorem.

We close this chapter with the statement of a problem. We say that a homotopy, H, of a differentiable manifold M into real N-dimensional affine space, RN, is regular of order p if, for each t E [0, 11, H, : M + RN is regular (= non-singular) of order p, and if the induced homotopy T,(M) x [O, l] -+ RN is continuous. We ask to classify immersions of M in RY up to p-regular homotopy. For p = 1 this problem has been studied by Hirsch and Smale [see, e.g., 121. That the higher-order problem is different from that of first order may be seen in the following way. By the Whitney-Graustein Theorem [22] the classes of first-order regular immersions of the circle S’ in R2 are in one-to-one correspondence with the integers, the correspondence being obtained by orienting both S’ and the unit circle in R2 and then assigning to each immersion f : S’ -P R2 the degree off, where p is the map which assigns to each x E S’ the positive unit tangent vector tofat x. The degree off is called the winding number off. But let f : S’ -+ R2 be an immersion with winding number zero, e.g. a figure- eight. Then, by a theorem of Sternberg and Swan [17], the Jacobian off vanishes at at least two points. But a point where the Jacobian ofj’vanishes is precisely an inflection point off. Consequently,

PROPOSITION 2.5. An immersion of rhe circle in the (real) plane with winding number zero has at feast two injlection points.

$111. THE HIGHER-ORDER GEOMETRY OF VECTOR BUNDLES

$111.1. The Differential Calculus of Vector Bundles.

Let V + M be a vector bundle over a manifold M. In this chapter we shall define, for each positive integer p, a new vector bundle API’, called the p-th derkatitje of V. For each p we shall establish an exact sequence of vector bundles over M

O+A,V-+A,+, V+o’+‘Ti@ V-40,

where 0 p+‘T, is the p-fold symmetric product of the tangent bundle of M. .-The main geometrical property of these constructions is described at the beginning of the next section (Theorem 3.12).

In spite of several attempts, we have not been able to reduce even the Ai construction to something familiar. However it may be helpful to the reader to remark that the initial problem was to find some vector bundle which stands in the same relation to an arbitrary vector bundle Vthat T2(the osculating bundle of order two) stands to T,(the tangent bundle). A,V is the bundle we want. To see this we show that there is always a homomorphism .s2 : AIT, + T2, embedded in diagram (3.17). In case M is a curve, &2 is an isomorphism.


Our plan of attack is to define A1 V, then to proceed by a recursion to the definition of A,Y. Everything is formulated and carried out for arbitrary coherent sheaves.

Let M be a manifold, ‘% the sheaf of germs of constant functions on M, 39 the sheaf of germs of (differentiable or holomorphic) functions on M, F1 the sheaf of germs of(differentiable or holomorphic) tangent vector fields on M (recall the conventions of $11.1).

Let Y- be a coherent sheaf of Smodules over M. Make Y @ (9-r 0 *Y) into a sheaf of 8-modules by prescribing the action f(a 0 D 0 b) = jiz OfD 0 b,f~ 9. Let .Br c 9’ @ (Y, @ .Y) be the subsheaf of &modules generated by all elements of the form - D(f)b 0 (D @fb - fD @ b),f~ 9. The action of 9 on V 0 (F-, @ wV) induces an action of S on YF @ F1 @ uYIW,-.

DEFINITION. The sheaf of Smodules %” Q (5, @ EV)/~,. will be called the first deriva- tiveofYunddenotedbyA,V. Letq: V” Q (.Fi Q %^l’) + A,V be the canonical projection.

Define a Shomomorphism I$- : 9’ + A,V by I&a) = q(a @ 0). Define a 9-homomorphism

(when we form tensor products of S-sheaves over 9 we shall omit the indication of 9 below the @), by P$-(a 0 D 63 b) = D @ b. It is easy to see that P;.(S?,) = 0, and hence P$ induces a 9?-homomorphism P,. : Ail’- + F 1 0 V. Let @i(V) denote the sequence

Let V and %‘lI be coherent !2-sheaves over a manifold M and let m : V + P‘ be a %homomorphism. Then we define a &homomorphism Aim : V” $ F1 B, cs^lr +

Yfi- @ Fl @ &- by A;m(a @ D ~$3 b) = m(u) @ D @I m(b). It is easy to check that A;m(.G%?,) c k&, so qA;m gives rise to a G&homomorphism A,m : A,V + A,Yf. We also get a map of sequences C,(m) : O,(V) + O,(W), which is to say that the diagram

is commutative. It is clear that Cf,(mn) = O,(m)@,(n), and E,(id.) = id. In summary

PROPOSITION 3.1. CEl is finctoriul; i.e., if m : Y -+ W is a 9!-homomorphism of k%- sheaces then m induces (I map ofsequences El(m) : cl(Y) + Gl(W).


LEMMA A. Suppose

0-A,---+A2--+

1 1 0-B,--+Bz--+

1 1 o-c,-cz-3

1 1 0 0

0

1 A,---+0

1

is a commutative diagram (of sheaves, etc.). Then 1. if the middle row and aN columns are exact, and if A, + A, --f 0 is exact, then C, + Cz -+ C3 -+ 0 is exact; 2. if the end columns and the middle row are exact and if Az + B, 3 C, + 0 is exact, then 0 -+ Cl + C, is exact; 3. if the columns and the top and bottom rows are exact, and if B, --) B, -+ B, is exact, then the middle row is exact.

The proof is a collection of standard diagram-chasing arguments.

LEMMA B. If 0 -+ “tr, --, “Y2 --* “Ir3 -+ 0 is an exact sequence of %homomorphisms of sheaves over a fixed base manifold, and if W is the sheaf of germs of sections of a vector bundle, then

is exact.

Proof: The lemma being local we may assume that Y#‘- r 9 0 . . . @ 9. Then the

sequence becomes

which is clearly exact, q.e.d.

PROPOSITION 3.2. el(Y”) is an exact sequence; consequently AIT is coherent.

PROPOSITION 3.3. QY1 is an exact functor on the category of coherent Q-sheaves over a fixed base manifold.

Proof of Propositions 3.2 and 3.3. It is clear from its definition that PV is onto and that

P,I, = 0. An element of the kernel of Pr is already of the form a @ 0 (mod .%?,), so it lies

in the image of I,. We must show that I, is injective.

Let YN be the sheaf of germs of sections of the product vector bundle of fibre-dimension

N, so that fN = 9 0 . . . 0 9 (N times). Let rn; : YN 0 2Tl 0 @SN + 9N be the Shomo-

morphism defined by w;(a 0 D @ b) = a @ D(b). Then w;( - DCf)b @ (D 8 fb - fD @ b)) = 0, so w;W#,) = 0, and Wi induces a Shomomorphism w, : A,jN + .YN, and w,INN =

identity, as one easily checks; so IS, is injective. w, is, in reality, nothing more than a

connection.


ml m2 Now let 0 4 ^yl -_) Y, + “Y, + 0 be an exact sequence of .%homomorphisms of

a-sheaves. It gives rise to a commutative diagram

0 0 0

1 Aimi 1 Km2 I 52 VI ’ %2 bL% f3 -0

4

I

4

A,% AiY,

+I

4

A1m2 AI+‘-,

,I A,V, -0

I I I 0 0 0

with exact columns. Now since W is a sheaf of fields,

O-,~~~V”lrl--)~_1~a.u,-,s,~.r,-tO is exact; hence the middle row of (3.1) is a direct sum of exact sequences, and so is exact.

Now Aim, : W,, --) &, is onto, for a generating element of BJVyj,

-D(f)b 8 (D @fb - fD 0 b), equals A;m2[- D(f )b ‘63 (D 6fb ‘-_!D 0 b’)]

where b’ is such that m,(b’) = b. Therefore, by Lemma A.l, the bottom row of (3.1) is exact at the middle.

Now let Y be a coherent sheaf of 9%modules; since the propositions to be proved are local we may assume the existence of an exact sequence of B-homomorphisms of .&heaves

ml m2

0+w49N+Y +o,

where jN is as above. This gives rise to a commutative diagram

I kv

nlll 1

0

P w I W-AIW----+ .F_1QW--+O

A1ml id. @ml

I f” PP I N O-9 ----+A19 N

-9~ @dJN-O

o--Y- A,lcr-----+F1 @Y-O

I 1 1 0 0 0


with exact columns (the right-hand column being exact by Lemma B and the middle by the last paragraph) and exact top and middle rows. Therefore, by Lemma A.2, the bottom row is exact, which completes the proof of Proposition 3.2.

Let us return td the exact sequence 0 -+ VI + Ypz + Yj + 0; it gives rise to a diagram

0 0 0

0 0 0

(the i-th column is just CZ1(Vi)) with exact columns; the top row is exact; the bottom row is exact, by Lemma B; the middle row is exact at the middle, as we have already shown; therefore, by Lemma A.3, the middle row is exact, and the proof of Proposition 3.3 is complete.

Let Y be a coherent sheaf of Q-modules; for each non-negative integer p we wish to

define a new sheaf A,T, called the p-th derivative of V. Let AOT = V; we have already defined AIY; let ~1~ : ArT + ArV be the identity map; we have already defined the exact sequence BI(V); let I, = I, and P, = P, (this slight irregularity of notation will permit simplification later). Assume that for each r, 1 < r < p, we have

l-r. defined a coherent g-sheaf A,T, 2-r. dejned a surjectioe Shomomorphism

pr : A,Ar-IV- + A/Y-,

3-r. defilled a Q-homomorphism P, such that the sequence

T, P, ~,(P^):O-,A~_,~~A~~-+O’~-,~Y-,O

is exact, where I,. = p,TAv_ ,,,., and 4-r. proved that

I4 44- ,^y b 4Y

(3.2,) Yr 1

P, al. I

Sl@O ‘-1yl @ *y‘- 0’9-1633, is commutatioe, where yr = (id. @ P, _ ,)Pa,_ ,v, and a, is the composition

DIFFERENTIAL GEOMETRY OF HIGHER OFDER 185

Sj being the i-th symmetrization operator. (One sees that o,(a @ b @ c) = a 0 b @ c.)

Our task is to carry out steps l-p + 1 to 4-p + 1. First consider the diagram with exact rows and columns

0 0 0

$(AP_ i?‘-) : O-A I

ql,) : I,

&(AP*Lr) : 0 - A !9 1

%&J : I’ I ,

1. I A,-,$< I P &-Iv I p-1 V--A A _ 1 9.’ V-- F1C3Ap-l -0

. 9 A,+--0

Alp, \ Yp+ I\

\ \

\ \

\

\ \

. \ . el(O PFl @ 9‘) : o-o v-1 8 % ‘-A1[Op~l@“y’]- WZ3OF

1 1 1 (3.3) 0 0 0

With pup left out it is COn'UnUtatiVe; yp+l iS @St the ‘filling in’. Let Y,+ r(v) = (IA&p - AII,,)(AiAP-lO; define xp+i Y = AiApV/Y,+i(Y) and let rp+r : AiA,Y + AP+tY be the canonical projection. One easily checks that yP+ 1(.9’p+ r(V)) = 0. Thus yP + 1 induces a Q-homomorphism yP+.i in a sequence

(3.4)

PROPOSITION 3.4. Sequence (3.4) is exact.

Proof. 7, + 1 is surjective, since it is induced by a surjection. The exactness at the middle is proved by a standard diagram-chasing argument. We must show that rp + lIA,p is injective.

Now IA,v~P(AIAP-IV n &Jp(A~Ap--l’f9 = IApvZp(Ap-,O = A~ZprAp_i&Ap-I~), for if

x is in the intersection, Ai P,(x) = 0, so x can be lifted uniquely under AlIp to y E Ai A, _ 1V; but PA,v(x) = 0, SO PA,_,v(y) = 0, ,and y can be lifted uniquely to Ap_17cr; so

~~#p(hAp-~‘f) n 4Ip(&Ap-1+‘-) = lA,V p Z (A,_,U). The inclusion the other way follows from the surjectivity of p,,.

1 B


Suppose w E APT with rp + II&~) = 0; then lift w under pLp to t’ E A1AP_r7cr. Now

lAPVpP(u) = lApY~P(u) - A,I,(u), for some 24, so AlIP = IAPVp’p(u - 21); hence 24 E IA,_,y(AP_r~) by the last paragraph. Let 1,$-,&u’) = U. Then

0 = lApVylPb’) - UPZAp_ ,vW

= zA,V~PzA, _ ,Y b’> - AIZPIA~ - ,v(U’> (by def. of IP)

= zApP~P(u)

= IAp&9;

but IA,V is injective, so w = 0, q.e.d.

Let g;+,(V) c (AP_rV 0 3-r 0 reAP_l F-) 0 .Fr 0 &AP-,V- 0 FI 0 aAP_,7’)= 9 P + 1 be the subsheaf of 9J4, + r g enerated over 9 by all elements of the form

O@E@(O@F@e)-O@F@(OOE@e)-O@[E,F]@(e@O),

where [ , ] denotes the Lie bracket of vector fields. Now the projection maps, q, give rise to a C&homomorphism q(q 0 id. @ q) : Q,+ 1 + AlAlA, _ r”Y-. Let 21, +1(V) = q(q 0 id. @I q)(2g+l(^lr)). The map pr, : AtAP-lV + A,Vinduces a map A,p, : A,A,A,_,%- 4 AIA,-Y. Let _c%?,+~(V’) = rpfl A&9’(V)). We have come to our main definition, l-p + 1; letA.,+,V = ~~+l~/~~+l(~)andlets,+,: ap+l”T + A,+,Vbethecanonical projection. Define p,,+r = ~~+~r,,+~ : A,A,V --t AP+l V”; pP+ 1 is a composition of sur- jections so is surjective, which completes step 2-p + 1. The situation may be summarized in the following commutative diagram :

A& rp+l A,A,A,_,V---- A,A,v B&IV

‘1 ‘1

(3.5) \

p”p+ A, Spfl

\ \ \

\ L I Ap+ I-Y-.

I P+l = pp + I IAPp, by definition. Thus we have a commutative diagram with exact

middle row and exact columns

P v P - ker(o,+ 1)- 0

i:_I” Y,+l 4

“-~~~“i’~~~l~+~~l~~~-O

E,+,(Y):O-A,-lr-A Y-Op+19-IO~~-0 P+l

1 1 1

0 0 0


(tl and p,+r are yet to be defined). The bottom row is to be the &,,+ 1 of step 3-p + 1.

a and /I are the inclusion maps.

PROFQSITION 3.5. Indiagram (3.6) ~p+l~P+,(kers,+I ) = 0; hence the diagram induces a map Pp+ i such that it remains commutative. With Pp+ 1 defined in this manner, diagram (3.2, + 1) is commutative, so step 4-p + 1 is carried out.

Proof. The second part of the proposition follows from the first, since diagram

(3.2, + I) is a subdiagram of

where the triangles are commutative, by definition of c(~+ I and y,,+l, and the square is a subdiagram of commutative diagram (3.6).

For the first half of the proposition we proceed as follows. By step 4-p we have the commutative diagram

(3.7)

Consider

AAA,-,~

(3.8) AIY~ &Pp


The left-hand square is commutative, since it is the derivative of the commutative diagram (3.7); the lower triangle of the right-hand square comes out of diagram (3.3), so it is commutative; the other triangle is commutative by definition of T,+I; hence (3.8) is commutative.

To establish the proposition it suffices to show that bP + Ijj, + lrP + ,A,&9b + 1(Y”)) = 0, which is the same as showing that

bp+1P~rg,0lAlb~A~1~~(Qb+l(ly’)) = ~p+~P~ps,cw A,q,A,y,dq 0 id. 8 qXQ;+ I(+?) = 0.

But

ap+ lP~D.~,~yAI~,AIy,dq 0 id. 8 4) .

x {08E8(O~F~~e)-O8F8(O8EOe)-O8CE,FlO(e80))

=CT p+l~~P~,~~~l~P~l(i~~ 8 P,-JAIPAp_,~q(id. @ 4)

~O0~~(O0F~~e)-O0~~~~0~8e)-O0C~,~lO(e00)~ =:d ,,+ IPop3r~fAIa,AI(id. @Pp,-Jq(O 0 E C3 F @ e - 0 0 F @ E 8 e} =U p+ lPopg,&lapq(O 0 E 0 F @ P,- de) - 0 0 F C3 E E3 P,- ,(4)

= cp+ lPops,ayq(O 0 E @ F. P,- l(4 - 0 0 F 0 E o Pp.- de))

= ~,+I(~8FoP,-,(4 - [email protected],-d4) = 0; this completes the proof.

Now that P,+l is in place in diagram (3.6) we have the mapping q at once, for let

-XE&J+, r; 0 = P,,lS,,, define q(x) = y.

a(x) = ~p+l~P+la(4r so Y,+I a(x) = /I(y) for some unique y;

PROPOSITION 3.6. q is onto; therefore, by Lemma A.1, I

P+l P

P+l

APV ---‘AP+lC-+~p+19-I @V----r is e.xact.

Proof. The kernel of bp + 1 is spanned by all elements of the form x = E @ F 0 Pp _ l(e)

- F 8 E 0 Pp_ l(e); but by the computation of the previous proposition x = ~p+l(~), where

Y = rP+ ,A,~,& 0 id. C3 q)(O 0 E @ P 0 F 0 4 - 0 0 F C3 (0 0 E 8 4 - 0 0 CE, Fl 0 (e 0 O)),

which lies in .$I,+ ,(V); hence q is onto, q.e.d.

Step 3-p -I- 1, and hence the definition of AP, for all p, will be completed as soon as we show that Ip + 1 is injective. The other steps have already been carried out. (The coherence of A,+, Y” follows from the exactness of CZp+l.)

PROPOSITION 3.7. (F, is finctorial, 1 < r < p + 1; more precisely, if m : V + W is

a C&homomorphism of S&sheaves, then m induces a 5%homomorphism A,.m : A,V + A,W such that the diagram


(F,(Y-) : o- k- ,Y - A,?- - (-J’LTI @F--O

e,(m) : A,-Iml A,m] id. @ ml

E,(W) : o---b Ar-@ - A,#” - o’Y~@w--o

is commutative; furthermore E,(mn) = C?,(m)C?,(n) and &,(id.) = id. If m is onto, so is Arm.

Proof. We have already established the proposition for r = 1 (Proposition 3.1). Assume the proposition for r, 1 < r < q < p, and, as an additional hypothesis, that the diagram

(3.9,)

is commutative, 1 < r < q; we shah prove it for.q + 1 and establish the commutativity of (3.9, + 1).

Consider diagram (3.3); replace p by q and call it !UI(Y). Using the functorial properties already established, together with the commutativity of (3.9,), one gets a map of diagrams (p’s included) %lJ(Y) --f %X(w). It is then clear that A,A,m(Y,+,(Y)) c Y, + i(W), (and equals 5~‘~ + ,(-w> if m is onto), so A,A,,m induces a &homomorphism 4 + lm : &, + lY

+ &, + lW. We also have a commutative diagram

4r+lm =

(A,_,m@id. @ A,_im)@id. @([email protected] A,-Im)

g,+ I(V)

I

) gq+ dW

&W,- Im I

A,&4--,+‘-

i

) AlAtip- ,W

AIAqm I AI A,v

I

,W,w a &+I~- q+lm I w

I ' q+l

A,.+,nl I A4+~~-_________________________________________,Aq+~~

(3.10)

where the middle square is just the derivative of (3.9,). One verifies by inspection that

6, + 1 42nq + 1 (VT)) c ii!: +i(W), (and, in fact, equals S;+i(-W) if m is onto), so a,+ ,m


induces a &homomorphism A0 + 1 m as required, and (3.10) remains commutative; diagram (3.9,+ 1) is a subdiagram of diagram (3.10) so is commutative.

The diagram I Q+l

4-y \ +A, \ /” \ \ (a) 1’ \

‘g.4 P /’

\4 h+1/

/ \ \ / \

L //

A1 A,$’

A,m (c) A1 A,m

I (d) A,+I’~

A1 A,Y~-

,/ ‘\\,

I 1’

I

*97/ (b) ‘\\Fq+ 1 \

/’ \

\ / I Y+l ‘L

1y

is commutative; for triangles (a) and (b) are commutative, by definition of I4 + 1, square (c) is commutative by Proposition 3.1, and square (d) comes from the commutative diagram (3.10); hence the left-hand square of diagram (E,, 1 (m) is commutative. We must show that the right-hand square is commutative.

Let 9(-V) denote the diagram (3.2, + ,). That the diagram

A,A,_,m

AlA,- I+‘- *AIA,_;$k

I

is commutative follows from the induction hypothesis; diagram (3.9,+1) is commutative, as we have just shown ; the diagram

id. ~3 tn @IF-- T-, @

1 id. @ tn 1 o”+lT1 c.3 f’ b Oq+‘fl 0 w


is clearly commutative; hence, since Z++ 1 is surjective, we have a map of diagrams g,(V) + r)(W) and hence that

P q+f A ~“-0 q+l

q+l F’,@V

I Aq+lm I id. @ m

P q+l I A W-0 q+i

q+l 9169.w

is commutative, which remained to be demonstrated.

Let I” be the product vector bundle of fibre dimension N and let IN be its sheaf of germs of sections. Recall that in the course of proving Propositions 3.2 and 3.3 we defined a map tnl : A,9N + 9N and showed that

PROPOSITION 3.8. ml defines a splitting of %,(YN); i.e., w~Z,N = identity.

Now we wish to define a map wq : Aq9N + sN, for all q < p + 1. Let m. : AoSN = .FN + sN be the identity. Suppose we have defined w, for each r, r < q, in such a way that the diagram

/4 AlA,_1.%N-A~4 N

(3.11,)

is commutative.

We wish to define wq + 1 and prove the commutativity .of diagram (3.1 I, + l).

PROPOSITION 3.9,. ZO~-~ = wqZq.

Proqfi

wq-1 = WII#NWq_ 1

= vU~~g-l~A,_~.w

= WqPqIAg _ , fN

= mqzq,

where the first equality follows from Proposition 3.8, the second from the commutativity of

1

Aq+YN Aq-,YN

) AiAq_14N

QJq-1

I

A.,m,- 1

1.0

YN : P A,$N

(which we have by Proposition 3.1), the third from the commutativity of diagram (3.11,) and the fourth by definition of Zq.


PROPOSITION 3.10. The diugrum

A A 9N-A.1§ N 1 P

7 \

/ \

\

Wq,”

\

/ WI’,

\

/’ \

Ai&_ p”/

\ L

(3.12) 9” \ 7

\

I&LA,

/’

ml/ /’

/ \

\ L Arm, /’

A A YN------+A,9 N/ 1 4

is commutatire.

Proof.

where the second equality follows from Proposition 3.9,, the third from Proposition 3.8, the fourth from the commutativity of diagram (3.1 l,), and the fifth from Proposition 3.1, q.e.d.

From Proposition 3.10 it follows that w,A~w,(Y,+~(~,)) = 0, so w,A,ru, induces a S&homomorphism UJ~ + r : & + r.YN + $N.

Now consider

A,A,A,_. ,YN

Triangle (a) is commutative, by definition of Pi+ I; triangle (b) is commutative, for

m,A,ro,A,p, = m,A,(ro,~J = w,A,(w,A,w,_,) = wIA,wlA,A,wq_I. By definition of

QJ;+~, $+,r,+r = wiA,w,; so diagram (3.13) is commutative.

DlFFERENTlAL GEOMETRY OF HIGHER ORDER 193

Now ~J,A,~,A,A,~,+ i(94+i(YN)) = 0. For

m,A,~,A,A,w,-,&I Q id. 69 4)

x (OQE~(O~FOe)-O~F~(O8E~e)-OQCE,~l~(e~0))

= q{O Q E 0 (hi,- de)) - 0 $ F 0 (Em,- de)) - 0 CD CE, Fl 69 q-d41

= EF(w,_ ,(e)) - FE(w,_ ,(e)) - [E, F](w,_ ,(e)) = 0.

So m; + ,(ker s4 + 1 ) = 0 and diagram (3.13) can be filled in with a map w*+ I such that (d) is commutative. Then triangle (c) is commutative, which gives the commutativity of (3.11, + i) and completes the definition of w4 + , , 1 < q < p.

We may now establish Proposition 3.9, + I ; the proof is the same as for Proposition 3.9, except q is replaced by p + 1, the only thing lacking to prove it before being the commutativity of (3.1 lp+ i). We conclude this section with

Z&homomorphisms correspond to vector bundles and vector- bundle homomorphisms. In the sequel we shall apply these constructions to vector bundles and we shall denote corresponding constructions by the same letters.

$111.2. Realization.

DEFINITION. Let V(“) --f M be a vector bundle of n over a manifold M and let cp : V(“) + FN be a map of V(“) into the afine space of dimension N (which we consider as a vector space) which maps each jibre homomorphically,


Let cp be a realization and let x E M. Let A,[cp(V(“‘/J] denote the linear span, in FN, ‘of all the osculating spaces of order q to the variety qp( Y(“‘) at the points of rp(P’I,). Let

AO[rp(V’“‘I,)] = cp(V’“‘I,). In the notation of $11 A,[cp(V’“‘jJ] = m,cp,(T,(V’“‘)~,).

Now consider the vector bundle FN --) {0}, where 0 E FN ,is the origin. Then if C, denotes the constant O-valued map on M, cp : V”) + FN is a homomorphism of vector bundles over C,,. Since FN + (0) is a product bundle, C6FN = IN, the product bundle of fibre-dimension iV over M. By $11.1 we have bundle homomorphisms c,, : CAFN = IN + FN over C,, and cp’ : Vtn) + IN, with c?& = cp.

THEOREM 3.12. Let V(“) --, M be a vector bundle offbre dimension n over a manifold M and let cp : V(“) -+ FN be a realization of V(“) in N-dimensional affine space. Then rp induces a canonical realization D,(q) = &u,A,cp’ : A,V’“) + FN such that for each x E M,

D,(cp)($ V’“‘j,) = 4&0’(“‘~.)1, P rovided, for the time being, that q G p.

Proof. Let U c M be a neighborhood of x which is a co-ordinate neighborhood with co-ordinates x1, . . . x,,, and such that V(“) is a product bundle over U, so there are n sections of V(“) over U S 1, . . . , S,, which are linearly independent at each point of U. The manifold V’“’ is paramitrized by x1, . . . , x,,,, a,, . . . , a,,, the a’s being fibre co-ordinates, so that cp can be written

Cp(X,, . . . , X,, al, . . . , a,) = i aiqSi(X,, . . . , X,). i=l

The osculating space of order q to cp at (x,, . . . , x,, a,, . . . , a,), is spanned by

for 1 < i < n, 1 < ,j, < m and 1 G r < q, (all other partials vanish) and hence the subspace of FN generated by all the osculating spaces of order q to cp along cp(V’“‘I,) is spanned by

(3.14) rPsitx), ax

for 1 < i G n, 1 < j, < m, 1 < r < p.

Now for all q = 0 there is nothing to prove. Let V be the sheaf of germs of sections of V(“). Then AIYIX is geneiatid, under the action of the structure sheaf 9, by allq(Yi 0 0) and q(0 @ ~/dXj @ 9’J, where .4pi is the germ of Si at x. But m,A,@i(Yi @ 0) = ‘~‘5“~ and

which establishes the theorem for q = 1.

Now suppose the theorem has been proved for q- 1. One sees from the form of the local expressions (3.14) that A1 [D4 _ ,(cp)(A, _ 1 V(“))l,] and A.,[& V(“)l,)] coincide. Hence

~,A~~,-,A,-,cp’)(A,A~-,V(“!I,)=6,[(p(V(”)J,)l.Butm,A,(~,-,A,-~rp’)= q~~(W,--,cp’) = w&A,&,, where the first equality follows from the commutativity of diagram (3.11,)


and the second equality from the commutativity of diagram (3.9,). Hence ZI,(cpXa,V(“)l,.) = A 4 [cp(Y(“)l )] q.e d X, *a

DEFINITION. We say that a realization cp : Y(“) + FN is non-singular of order p at x E M if the realization D,(q) : A,V’“) + FN is non-singular (of order zero) at x.

PROPOSI~ON 3.13. A,V’“’ is just the right size; i.e., if x E M is an arbitrary point, there

exists a neighborhood U of x such that V(“‘lo admits a realization which is non-singular of

order p.

Proof. Let U be a co-ordinate neighborhood of x over which V(“) is a product bundle, so we have an isomorphism i : V’“‘I, + I”.

Let s be a section of I’, and let f: U + FL be a map of U into an affine space FL which is non-singular of order p. (The existence of such a map is guaranteed by Theorem 2.4.)

Define cp” : I’ -+ FL x I;l = F=+l b y cp”as((x)) = (af (x), aV), where V # 0 is some fixed vector in F’. Then the osculating space of orderp to rp” at as(x) is of dimension 1 + v(m, p).

Now define cp’ : I” + FL+’ x FL” x . . . x FL+’ = F”(=+l) by cp’ = rp” x cp” x . . . x cp”, and define cp : V(“) + FntL+t) by cp = cp’i. Then the osculating spaces of cp are clearly of dimension n(1 + v (m, p)), which is the dimension of A,V’“‘; hence, by Theorem 3.12, 40 is a realization which is non-singular of order p, q.e.d.

I P+l

PROPOSITION 3.14. 0 --t ApIN -+ Ap+lZN is exact.

Proof. The proposition is local, so let cp : IN -+ FK be a realization of IN which is non- singular of order p. Then if cp’ : I” + ZK is the induced homomorphism of sheaves, wpApr++ is injective. Now the diagram

Z P+l

Ap.FN - A

I

P +

Ap@ I_ A,+ 1’~’ I P+l I

A/.-A 3” P+l

\ I \

WP\ wp+1 \

\ \

\ \

L 9K

is commutative, (the square by Proposition 3.7 and the Hence Z, + r : API” + A, + 1 IN is injective, q.e.d.

I P+l

PROPOSITION 3.15. Zf V is an arbitrary coherent g-sheaf, 0 + A,Y - Ap + 1 V is exact.

This completes step 3-p + 1 of the preceding section; hence the definition of A,Y for all p is completed and a/l our propositions are established for arbitrary order.

triangle by Proposition 3.9, + r).


PROPOSITION 3.16. $+I is an exact functor on the category of coherent Sksheaces ocer a fixed base manifold, p > 0.

Proof of Propositions 3.15 and 3.16. We have already established Proposition 3.16 for

P = 0; assume it for r < p. Let 0 + Y, + Y, + V, + 0 be an exact sequence of .Shomomorphisms of Q-sheaves over a fixed base. Then we get a commutative diagram

Ar+ I+‘-t - Ap+r~t- Al+~“lr,-0

I I 0 0 0

with exact middle row and exact columns. Now CL is onto, as one can see from the proof of Proposition 3.7. Hence by Lemma A.1, the bottom row is exact.

The propositions to be proved are local; so we may assume, by the coherence of V, that we have an exact sequence 0 + -Hr + 4N -+ V + 0 of z&homomorphisms. Then we

get a commutative diagram

0 0

I I 0 - A;f’-

I ‘+I Ap,l~----+

I O”f?-t Q -lk”- 0

Z PC1

I I 0- A#” --, Ap+ 19N- op+%-r QYN---+0

I I I I 0- A,Y- ‘+r ApclY--+ op+Q-IoY-o

I I I . 0 0 0

with exact columns (the middle cohunn is exact by the previous paragraph) and middle row. Thus, by Lemma A.2, the bottom row is exact, which demonstrates Proposition 3.15.


Now we have that the diagram

0 0 0

I I O-----+A,Y1 ,A “Irz

i

+ A,V”, - 0

1 I 0-A VI-----

P+l + A,+ IYZ

I 1

---------*A “y3-O( P+l

I

O---+OP+lF-_lO* 1-o p+cq @ vy2---+ op+QT1 @V3--+0

i I I

0 0 0

is commutative with exact columns and top and bottom rows, and with the middle row exact at the middle. Hence by Lemma A.3 the middle row is exact, which completes the

proof of Proposition 3.16.

3III.3. Schoiium.

We begin by recalling some notions from [2]. An exact sequence

of vector bundles over a manifold h4 is called an extension of E” by E’. Two extensions E,, CZz of E” by E’ are called equivalent if there exists an isomorphism of one onto the

other of the form

@i : 0---rE’---rEl---+E”-0

id.

I 1 I

id.

4 (5, : O- E’--+EZ-E”-0.

It is shown that the set of equivalence classes of extensions of E” by E’ is in one to one correspondence with H’(M;&‘om(E”, E’)), the first cohomology group of M with coefficients in the sheaf of germs of homomorphisms of E” into E’, with the split extension corresponding to the zero element. The cohomology class corresponding to an extension is called the obstruction to the splitting of the extension.

The correspondence is obtained as follows: let (5 be an extension and let 6* : H”(M; Xm(E”, E”)) + H’(M; .I%/H(E”, E’)) be the coboundary with respect to the exact sequence of sheaves &~r)z(E”, (5) : 0 + 2mz(E”, E’) -+ SW~(E”, E) + &&(E”, E”) + 0; then if I E H’(M; Xm(E’, E”)) is the cross section of &‘&E”, E”) which gives

everywhere the identity map, 6*(Z) corresponds to the extension CL?.

PROPOSITION 3.17. Let V(“) be a rector bundle over a manifold M which is given, with respect to a covering {Vi}, by isomorphisms of vector bundles Ui : I-“lu, + V(“$,,, where I” is

198 WILLIAM FRANCIS FQHL

the product vector bundle offibre dimension n and Iv, denotes restriction to Ui. Letgij be the transition functions of V (“‘.

= uj-‘uj Then the obstruction to splitting the extension

c&( I/‘“‘) : 0 -+ V(“) + A, V(“)+ TI @ V’“‘+ 0 f

d( I’(“)) E Zf’(M; %GZ(T, @ I’(“), V’“‘)), is represented by the cocycle 6, defined by 6,(D Ob) = -utgtjD(gjJuY’(b)+

PY Proof. Consider LEI(-lr) : 0 + Y + A,Y + F1 @ Y + 0, where Y is the sheaf of

germs of sections of V’“‘. Over each Ui define ai : Fl @ V + Al-Y- by di(D @ 6) = q[-uiDui-‘(b) @ D @I b]. ClearlyP,6i = identity. ai is a $&homomorphism, for

ai(D Ofb)

= q[-uiDui_‘(jb) @ D @,fb]

= q[(D(f)b - LltfDu;l(b) - utu;lD(f)b) @jD @ 61

= Si(fD 0 b).

The obstruction to splitting CE1(VCn)) is represented by aij = Sj - ai. But

(Sj - ~i)(O 0 b)

= (-ujDu;’ + u,Du;‘)(b)

= (- “igijD(gji)u; ’ - UigijgjiDU; ’ + u~Du; l)(b)

= ( - UigijD(gjJuT l)(b), q.e.d.

Now Atiyah also has a method for constructing new bundles from old in a differential, fashion; in fact, our theory is modeled on his. Given a vector bundle V(“) he defines a new vector bundle D(V’“‘) and an exact sequence

B(V) : 0 --f Tr @ V(“) + D( I/‘“‘) + I/‘“’ + 0,

where * denotes the dual of a vector bundle. The relation between D and AI is given by

PROP~XTION 3.18. Let b(V(“)*)* be the obstruction to the splitting of

a( V(“)*)* : 0 -+ I/(“) + D( V-c”‘*)* + Tl @ V (“‘4 0,

and let d(W)) be the obstruction to splitting CEl( V’“‘). Then d( I’(“)) = - b( V(“)*)*.

Proof: According to [2], the obstruction to splitting 99(V0))) is represented by the cocycle bt (I’(“)) = -ui(dg..)g..~; ‘.

by the isomorphisms UT - ’ Now V(“)* is given, with respect ‘to the covering {Vi}

:)+fi,, + p)*l “, and the transition functions of V(“)* are uz?uT -I L gTi. b( V(“)*), the obstruction to splitting B( V (“)*), is then represented by - uz? -‘(dg~&~,?~ut. Then, by Proposition 3 of [2], b(V(“)*)* is represented by Pij, where

pij(D @ c) = (u;- lD(gfi)g:ju;)*(c)

= uigijD(gjt)u; ‘(c)s q.e.d.

PROPOSITION 3.19. Let A4 be a compact Kahler manifold, V(“’ + M an analytic vector

bundle, and L(V(“)) the vector bundle associated to V(“’ under the adjoint representation of

thegeneral linear group. Then, under the canonical isomorphism H’(M;XWJ(T~ @ V(“), I’(“)))


+ H’(M; L(V(“)) @J T*) [2, Proposition 91, d(V(“)) goes into an element which generates the Chern characteristic ring of V(“’ under the operation of the invariant polynomials of the

general linear group.

Proof This is a combination of our Proposition 3.18 with Theorems 3 and 5 of [2].

So if V(“) is a complex-analytic vector bundle over a complex manifold, Ei( V(“)) splits only if all the Chern classes of V(“) vanish. (The converse is not true, as is shown by an

example in [2].)

ApV(“) is a filtered vector bundle V(“) c Ai V@) t AZ@“) c . . . c A,,V’“’ and its transition functions depend on the p-th order partial derivatives of the transition functions of V(“). For example the transition functions of A3 V(“) are of the form

’ I/ii Aij Dij f’ij ’

0 J~jO I/ij Bij Eij >

0 0 Ji’j 0 tij Cij

,O 0 0 Jt 0 V,,i,

where Vii are the transition functions of V(“); JTj @I Vij are the transition functions of

0’7, @ V’“‘* A,, B,,, Cij are functions involving the first partial derivatives of V,; R,, E, are functions involving the second partial derivatives of Vij; and Fij are functions

involving the third partial derivatives of the I’,.

I_et us next examine some relations between $11 and §III. Let T4 be the bundle of

q-th order osculating vectors of M.

PROPOSITION 3.20. For each p > 1 there is a vector-bundle homomorphism .ep : AITp- 1

+ Tr which is surjective, and such that

I T,-I P T,-1 E,(T,_,) :0-7&-A T _

1 P 1 -T, Q Tp_l-O

(3.15) id.

I

EP EP

1, I ,I PP

2,(M) : 0 -4 T,- , --- T D OUT, -0

is commutative, where lower sequence Is that described in Theorem 2.1 and E; is the composition

id. @ P,_, id. Q S,_, SD Tl 0 Tp-l --TI o”-‘T,--- @!“T, + OPT,.


I Proof. We do everything for the corresponding sheaves. Define

$:9--,-l 89-l oyF’p-l + F,, by $,(D Cl3 E Q F) = D + EF. Then E;(&?~-,_ ,) = 0, for

s;(D 8 E OfF - (D f E(f)F) OfE @ F) = D + EfF - D - E(f)F - fEF = 0,

so 8;: determines a %homomorphism .sp as required. The commutativity of the left-hand square of diagram (3.15) is immediate; that of the right-hand square is established when we show that e;Pg,_,q( D @ E @ F) = PJEF). But

s;P~,_,q(ZXMOF) =s;(EOF)

= E OP,_,(F)

= P,(EF), q.e.d.

It is also possibIe to define a map sp,q of APT, onto Tp +*, but we shall not go into details for we shall not need it here.

PROPOSITION 3.21. Let t,(M) be the obstruction to the splitting of

0+T,-+T,+T,~T,-+0.

Then &&(S2)*(d(Tl)) = t,(M), where d(T,) is the obstruction to the splitting of El(T,)

and &%~@Z)* : H’(M; .%&/(Tl @ Tl, T,)) + H’(M; Xua~(T, 0 T,, T,))

is induced by the symmetrization map.

Proof. In case p = 2, diagram (3.15) becomes

O-‘T,--+A,T,----+ Ti @ T,-0

(3.16) id.1 %I $1

0-T,----+T,----+ T,OT,- 0.

Consider the commutative diagram s’

Zf’(M; %cwz(T,, T,)) - H’(M; .%~t(T, 0 T,, T,))

id.

I

&&(S,)’

1 H”(M; .%xz(Tl, T,)):: H’(M; J&,L(T, @I T,, T,))

which arises from diagram (3.16), where 6 *, P’ are the coboundary homomorphisms. Then if Z is the identity cross-section one can show, by analogy to Exercise 1, p. 308, of [4], that 6*‘(Z) = -d(T,) and 6*(Z) = -t,(M). Hence -$&~(S,)*t,(M) = -d(T,). But remember, S2 also goes the other way, and the composition

.3%&S,)*

------+H’(M; &nl(T, 0 T,, T,))

is the identity. Hence tl(M) = &&(S2)*(d(Tl)), q.e.d.

DIFFERENTIAL GECMETRY OF HIGHER

THEOREM 3.22. Zf M is a compact Kiihler manifold and

O~T,+T,+TIOTI+O

splits, then all the Chern classes of A4 are zero.

ORDER 201

Proof. By Proposition 3.19 it suffices to prove, under the hypothesis of the theorem, that

O+T,+A,TI-+TI~TI-+O

splits. Now we have a commutative diagram

0 0.

O-ker EZ--+TI A TI-30

I I A2i (3.17)

with exact rows and columns, where A, is the alternation map and 8 is induced by the mat of the diagram. If m : T2 + TI is a splitting of the bottom row, then ru~~ : AIT, -t T, is a

splitting of the middle row, q.e.d.

9ItI.4. Some Preiiminaries.

This section is independent of everything which has gone before.

PROPOSITION 3.23. Let V(“) be a complex (resp. algebraic) vector bundle of$bre dimension

n over a paracompact space (resp. non-singular algebraic variety). Let cXV(“)) denote the i-th Chern class of V(“). Zf x is an indeterminate and

$.. ciCv(“))xi = ,fil ( l + Yjx) is a formal.factorization, then

i~c Ci( 0 pv(n))Xi = n ISi,< . . . <i,Cn

[l + (Yi, + Yi* + **a + Yi,)x3.

The proof for the topological case is quite analogous to that for the exterior algebra, and we refer the reader to [13]. The algebraic case follows at once, since the techniques of [13] are also valid in the algebraic case [l I].

PROPOSITION 3.24. Let I/(“) 4 M be real vector bundle of $bre dimension n over a

paracompact space M. Zf wi( V(“)) denotes the i-th Stiefel- Whitney class of V(“) and

ii wi( v’“‘)xi = ,fil C1 + Yjx) T c C


is a formal factorization, with x an indeterminate, then

iio Wi(OPVnpi = n 14i16 . . Qi,Qn Cl + (Yi, + ... + YiJxl,

This may be proved using the methods of [18].

Let M be a complex manifold and let V and W be complex line bundles over A4 which are given, with respect to a covering { Ui} of M,, by transition functionsfij, gij, respectively. Let f: V +’ W be an analytic homomorphism of line bundles. Then we can choose fibre co-ordinates t’i, Wi, in VI”,, W(u, SO that Ui = fijUj and wi = gijwj; andf(ui will be given by wi = (pi(x)ui, where qi(x) is a holomorphic function defined in Ui. The ‘pij = cpi/qj = gijhi

are holomorphic, non-zero functions which determine a divisor D(J), called the divisor of f, provided f does not identically vanish in any open set. The following is immediate.

THEOREM 3.25. (The Riemann-Hurwitz Formula.) Let f : V + W be a complex-analytic

homomorphism of complex line bundles over a complex manifold which is not identically zero

on any component of M. Then LDcf), the line bundle associated to the divisor off, is analytical@

isomorphic to V* Q W, where * denotes the dual of a vector bundle.

COROLLARY 1. Let

f y(n) - W -(a)

I I f” M-----,N

be an analytic homomorphism of complex-analytic vector bundles of jibre dimension n over an

analytic map of complex manifolds M f” -+ N which has maximum rank somewhere on each

component of M. Then if we define the divisor of J D( f ), to be the divisor of the line bundle

homomorphism A’f: A’V’“) -+ A’( f “! WC”‘), where A” denotes exterior product, c,(LD( f)) = f “*cl(W(“)) - cl(Vcn)), where c1 denotes thefirst Chern class.

COROLLARY 2. Let g : M” --, N” be a complex-analytic map of n-dimensional complex

manifolds whose Jacobian is somewhere of maximum rank on each component of M”. Then if D(g) denotes the divisor of A”T,(M”) + A”g!T,(N”),

c,(LD(g)) = g*c,(N”) - q(A4”).

The original Riemann-Hurwitz formula is Corollary 2, with n = 1.

§IV. THE HIGHER-ORDER GEOMETRY OF SUBMANIFOLDS

pIV.1. Submanifolds of Grassmann Manifolds and Associated Maps.

Let G,,, be the Grassmann manifold of all a-planes through the origin of an a + b- n

dimensional affine space, F“+*. Let Eel,* + G,,, be the universal a-plane bundle over G,,,.

We may think of EOab as consisting of all pairs (P, u), where P E GO,* is an a-plane through the origin of F”* and v is a vector in P. The projection map TC is defined by 7c(P, U) = P. Let

= 1 E.,b + F”+* be the map (P, u) + v. Then c is a realization of the vector bundle Eo,* 3

G o,b, in the sense of the previous chapter.


Now letf: M -+ G,,, be a map of a manifold into a Grassmann manifold. f !E,,* denotes the inverse image, under f, of the universal bundle Elr,b. Then we have a (non-singular) vector bundle homomorphism

P

J’k. b ---+&.b

wberepis induced by f (§ 11.1). Clearly the map of is a realization off’E,,b in the sense of the previous chapter; by Theorem 3.12 it induces a realization D,(af) : fipf*Ea,, + Fa+h, where AP denotes the p-th derivative of a vector bundle.

DEFINITION. Wesay that the mapf: M + Go,b is non-singular of order p at x E M if the

realization D&o-) of A,f!E,,, is non-singular at x, i.e. if the restriction of D,(a-) to thejibre

over x is injective.

DEFINITION. Letf : Mm + Ga,b beamapofanm-dimensionalmar@oIdMm intoaGrassmann

manifold, GaVb, which is non-singular of order p. The map ftp) : M” -+ Gi,fl+b_j (where

J ’ = a(1 + v(m, p)), v(m, p) being the number defined in SII) defined by f@“(x) =

D,(d) (A,f!&,lA h w ere Apf!ElrSblx denores the fibre of L\pf’Ea,b over x E Mm, we call

the p-th associated map to f:

&qpOSe f: Mm + Ga,b is a map of an m-dimensional manifold Mm into a Grassmann manifold. Then, for a given integer p ‘%- 0, .f falls into one of the following cases:

1. f is non-singular of order p everywhere;

2(a). f is non-singular of order p almost everywhere and the p-th associated map to

f ((Mm - S,), where S,, c Mm is the set of singular points off of order p, can be extended analyticaliy to all of Mm. In this case we still speak of the p-th associated map and use the same notation, f (p);

2(b). f is non-singular of order p almost everywhere, but the p-th associated map to

f (CM” - S,) cannot be extended over all of Mm;

3. f is everywhere singular of order p.

Suppose f falls into case 1 or 2(a); then we ask for the homology type off(P). To give

this it suffices to determine the characteristic classes off (p)!Ej,a +b_j

THEOREM 4.1. Let p be a positive integer and let f : M” + Gaeb be a map of a manifold Mm into a Grassmann manifold Go,h, which falls into case 1 or 2(a) above. Let f (p) : Mm

+ Gj o+b-j be the p-th associated map. a,f: hpf!EYvb -+ Ej.,+b-j OlVr f”‘,

Then f determines a homomorphism of vector bundles which is an isomorphism wheretier f is non-sing&r of

order p.

Prooj: The required map, ?,f, is defined iy a,j(x) = (J-(‘%(X), D,(a-)(x)), where x is

the projection map of the vector bund1e.f !E,,b --* M”. Clearly a,_/ is non-singular if and only if D&of) is non-singular; but this is precisely the condition that f is non-singular of orderp.


COROLLARY 1. Let f be as in the theorem and non-singular of order p. Thenj.‘P’!Ej,U +b _

is topologically isomorphic to

f&b 8 (f @ i$lo~T,(iM”)),

where x and Q denote Whitney sum, Oi the i-fold symmetric product. T, the tangent bundle

oj’ M, and I the product line bundle.

COROLLARY 2. Let f be as in Corollary I. !f ci denotes the i-th Chern class then.

in Ci(f!E,, *)Xi = Ip (l + WjX)

j=l

are formal factorizations defining the y’s and w’s and x is an indeterminate.

COROLLARY 3. Let f: M,,, + G.,b be a complex-analytic map of a complex manifold M,,

of dimension m into a Grassmann manifold which is non-singular of order p almost everywhere

and such that f (P’ is defined (case I or Z(a) above). Then

c,(f'P'!Ej a+b-j) = 0~ -(vh P) + 1)~ + aA4 pk,(W,

where ap is the Chern class of the divisor of a,,f (see §III.4), o = f “(f-l), where R is the funda-

mental two-class of Gu,h, and

P

Cc(m, p) = I( m+,‘-1 V(M, pj =

m+r-1

r=1 I‘- i 1 ’ r&(# r ,I* -

Proof @‘Corollary 3. By the Riemann-Hurwitz Formula (Theorem 3.25)

c,(f (p’!Ej, o+h_j) = UP + Cl(Apf !E, b)

But

@,_%,J = Cl ‘k, b (, - 0 (10 ,~lO’T,(Mm))) = -44~ P) + 1)~ + aAm, p)c,(M,),

q.e.d.

(This result appears to be new even if a = p = 1.)

If f is as in Corollary 3 there are universal polynomials which relate the higher Chern classes of the bundles concerned with the singularities, according to the theory of Thorn and Haefliger. (Thorn considers a map f: Mm+’ + Mm, r > 0, of complex manifolds which is not totally singular, and shows that there are universal polynomials which relate the Chern classes of Mm+‘, M”, and the singularities [ 191. Haefliger considers a homomorphism of vector bundles which is not totally singular, f: Vcm+‘) + Vcm), over a fixed base space and shows that the same polynomials relate the Chern classes of V(m+r), Ycm’, and the singularities off. Haefliger’s generalization was given in a short communication to the International Congress of Mathematicians, 1958.) These polynomials have recently been

computed by 1. R. Porteous.


6IV.2. Submanifolds of Projective Spaces.

G .Nv the Grassmann manifold of lines through the origin of N + l-dimensional alTine space, is also called N-dimensional projective space, P,.

Let f: M --) PN be a map of a manifold A4 into a projective space. Considering PN as a Grassmann manifold, we already have the notions of non-singularity of order p offand of thep-th associated maps. However in this case the associated maps have a simple geometric interpretation. Let t : ( FN + ’ - {O>) --f PN be the map which assigns to each point of N + l-dimensional affine space other than the origin the line joining it to the origin.

DEFINITION. ~D,(o--)(A,,j”!E, ,& - (0)) we calf the osculating projective space of order p to f : M + P, at x. The osculating projective space of order I, of course, is called the ta.vgent

projective space.

Using the realization theorem (Theorem 3.12) it is easy to show that this definition agrees with the usual definition. Our singularities of order one are precisely the singularities in the usual sense, i.e. points where the rank of the Jacobian off falls. Our singularities of order two, however, are precisely the inflection points.

Now the Grassmann manifold, G+ of a-planes through the origin of affine a + b-space may also be regarded as the manifold of projective a - l-spaces in projective a + 6 - l- space, the correspondence being given by the map r; i.e., if P is an a-plane through the origin of afine a + b-space, T(P - (0)) is th e corresponding a - l-dimensional projective subspace. Hence the p-th associated map may be regarded as the map which assigns to each x E M the osculating projective space of order p to f: A4 -+ PN at x.

Now if p is a positive integer f falls into one of the cases discussed in the previous section. If f is everywhere non-singular of order p, or, what is the same, if the osculating projective space of order p to f: M + P, is everywhere of maximum dimension, then the homology type of the p-th associated map is given by Corollary 2 of Theorem 4.1, with a = 1. The first-order case has been known for some time; it is called the Todd formula, and it is historically important because its discovery first established the correspondence of the Todd canonical systems of an algebraic variety to the Chern classes [15].

The existence of maps into projective space which are non-singular of orderp is guaranteed by the following

THEOREM 4.2. Let p be a positicle integer and let M be a manifold which admits a map

(resp. a:1 e nbedding) into a projectice space which is non-singular in the first order. Then M

admits a map (resp. an embedding) into a prqjectiue space of suficiently high dimension which is non-singular qf order p.

Proof. We prove the theorem first for M = PN, N-dimensional projective space. List all (unordered) p-tuples of integers [iI, . . . , i,,], 0 < ij < N, as F,, . . . . F, and define, for each k, Fk : FN+’ + F’ by F~(x,, ~ . . , xN) = Xi,Xi~ . . . xip, where [il, . . . , ip] is the k-th p-tuple, and where the x’s are the co-ordinates. Then the Cartesian product of the F,‘s

defines a map F” : FN+’ + FL+’ . Now F” is homogeneous of degree p; therefore it defines


a map F: P,Y -+ P,, and, just as in the affine case (Theorem 2.4), one shows that F is non- singular of orderp and an embedding. Thus D,(aP) is a non-singular realization of A,F!E, ,L, where fi : F!EI., + E,,, is induced by F.

Now let ii4 be a manifold which admits a mapf: M + PN which is non-singular of order one. Then Ff: M + P, is non-singular of order p as required. Iff is an embedding, since F is an embedding, so is Ff, q.e.d.

The following question still seems to be open: if M admits a map into a Grassmann manifold, Gp,b, which is non-singular in the first order, does it admit a map into a Grassmann manifold Gs.L, for large enough L, which is non-singular of order p?

Let p be a positive integer and let f : M + P, be a map of a manifold M into a projective space which falls into case 2(a) of the preceding section, i.e., f is non-singular of order p almost everywhere and the p-th associated map is defined even at singular points. Then we have the vector bundle homomorphism

3J: AJ!EI,N + Ej, t+N-j, (j = v(m, P) + 1)

described in Theorem 4.1 and the singularities of a,f we call the p-th-order singularities of f, those of first order are singularities in the usual sense, those of second order are called inflection points. Again, by the theory of Thorn and Haefliger, there are, in the complex- analytic case, universal polynomials which relate the Chern classes of the two bundles with the cohomology classes dual to the singular subvarieties. We shall show how all this works out in the case of a curve, where we shall get numerical results.

LEMMA 4.3. Let V(“’ + M be an analytic n-plane bundle over a curve M and let p : V(“’ + FN be an analytic realization of V(“) in FN which is non-singular almost everywhere, so that the singular points form a discrete set of points S c M. Then the map p’ : (M - S) + G,,, _-n

defined by p’(x) = p(V1,) can be extended analytically over S.

Proof: Gn,N_n may be regarded as the set of equivalence classes of non-zero decomposable exterior n-vectors, two such, Q, /?, being called equivalent if CL = afl where a is a scalar. Let x E M be a singular point of p and let U be some co-ordinate neighborhood of x, with co-ordinate z, z(x) = 0, with x the only singular point in U, and over which V(“) is a product bundle, so that there are n sections of V(“) over U, S,, . . . , S,, which generate the fibre of V(“) at each point. Then if y E (U - {x)), p’(y) . IS re p resented by the decomposable n-form A(y) = pSr(y) A . . . A&?,(Y), and this n-form vanishes only at x. Therefore

A(Y) = zqR( y), for some integer q and some decomposable n-form n(y) defined and non-zero throughout U. But A is equivalent to R where this is meaningful. Therefore we may take Q as the form defining p’ and get the desired extension of p’ to all of U. Since x was an arbitrary singular point the lemma is proved.

This lemma amounts to saying that maps of curves into Grassmann manifolds which fall into case 2 of the preceding section fall into case 2(a).

Let us now restrict ourselves to the complex-analytic case. Let g : M + G,., be a complex- analytic map of a closed complex manifold M,, of dimension 1, into a Grassmann manifold. Then g induces a map g* : H,(M, ; 2) + H,(G,,,; 2) on the integral homology. There is


only one Schubert cycle of (real) dimension 2, call it /I; let c1 be the fundamental (real) 2-cycle of MI. Then g*(a) = v(g)p for some integer v(g) which we call the order ofg. LA

us agree to adopt Hirzebruch’s definition of the Chern classes 1131. Then v(g) = -rc[cl(g!E,,J], where c1 is the first Chern class and K denotes evaluation of a cohomology class on u. The order can, using elementary facts of Klhler geometry, be interpreted as the ‘arc length’ of the curve g : M + G,,,.

Now let f: M + PN be a map of a complex-analytic manifold of dimension one into N-complex-dimensional complex projective space which is not everywhere singular of order r in the sense of the previous section. Then, for each p < r, f falls into case 1 or 2 of the previous section, hence into case 1 or 2(a), and so we have the p-th associated maps fcp): M, + G,,, n-p. Let v,, be the order off, vp the order offtp), dp the degree of the divisor of a,f, i.e. rc[c,(tD(~$))J, where LD denotes the line bundle associated to the divisor of a homomorphism of vector bundles, and d,f : AF f !E, ,” + Ep + 1 ,” _p is given by Theorem 4. I. We call d, the total singularity of orderp ofjI The second difference of the d’s,

wp = d ,,+I -2d,,+d,-1,

we call the p-th stationary index off. (The geometrical significance of wp is discussed in [21].) Let x(M,) = K[c~(M)] be the Euler-PoincarC characteristic of M. We get the following classical theorem :

THEOREM 4.4. The Pliicker Formulas.

-wp - vp-1 + 3, - VP+1 = X(M,), O<p<r, (v_~=O).

ProoS. By Corollary 3 of Theorem 4.1

d, = +W’(J,J~)l = -VP + (p + l)v, - F X(&f,)

Hence -wp - \‘,+I + 2v, - VP+1

= -d,+, + 2d, - d,_, - vp-t + 2v,- v,+1

=vp+t -(P+W,+ 2

(P + U(p + 2) X(M ) 1

- 2vp + 2(P + lh - P(P + 1)XWl)

+ vp-1 - pvO + ” - ‘) x(llJ ) 2 1

- VP_1 + 2v, - v*+t

= X(M,)* q.e.d.

Suppose f : A4, --t PN is complex-analytic map of a higher-dimensional complex manifold into a projective space which falls into case 2(b) of the preceding section. Then, under certain assumptions on the singular subvarity S, it is possible to ‘blow up’ h4, to obtain a new manifold A,,, and a map fl,,, 4 M,,, in such a way that the map (f ‘((M,,, - S)n)‘P’ extends to a map 3(P) : A, + Gj,, -j, which reduces case 2(b) to case 2(a), and leads to relations between the intrinsic invariants of M, and the invariants of the map f. This approach is studied, for the first-order case, in a forthcoming paper of Harold Levine.

208 WILLIAM

We consider next some instances of maps of complex manifolds into projective spaces which fall under case 3 of the preceding section, i.e. which are everywhere singular of order p.

Take a ruled surface in a projective space. If it is developable its osculating spaces of order 2 are of dimension 3 in general. If it is a scroll, i.e., is non-developable, its osculating spaces of order 2 are of dimension 4 in general, whereas the osculating spaces of order 2 of a surface which is non-singular in the second order are of dimension 5.

But a ruled surface may also be considered as a curve in a Grassmannian of projective lines in a projective space and conversely. So let f: M1 + G.,b be a complex-analytic map of a complex curve M, into a Grassmann manifold which is not everywhere singular of order I’. By the total singularity of order p off, dp, p < r, we mean the degree of the divisor of

a,f: Agf ‘E,.b + Ej,P +b--j, j = a + ap. If we now regard f: M, 4 G,,, not as a curve in a Grassmann manifold, but as a one-parameter family of a - l-planes, called generators, in a+b - l-dimensional projective space, then if x E M,, f @j(x) may be regarded as the linear span, in P. +b _ I, of the osculating spaces of order p at the points on the generator corresponding to x. A singular generator is one where the dimension of this space falls. Let v,, be the order offCp) and vO the order off.

THEOREM 4.5.

d, = -vp+(p+l)v,-a~ p(p + l) @f,) 9 0 < p < I’.

Proof. Immediate consequence of Corollary 3 of Theorem 4.1.

A developable surface f : Ml + G2 ,N _ 1 is totally singular in the second order.

PROPOSITION 4.6. Let f: M, + PN be an analytic curve in a complex projective space which is not everywhere singular of order r + 1: consider its tangent developable, V. Iff (P) is the map which assigns to each generator of V the osculating space qf order p at that generator (the osculating spaces are constant along the generator), vp = v(f (P)), v,, = v(fl, and d, the total singularity of order p off, then

d,= -v,+(p+2)v,- (P + l)(p + 2) i((lM )

2 1 *

Proof. Let the curve f be given locally by a vector function X(t) in N + l-dimensional affine space. Then the tangent developable is given by the one-parameter family of 2-planes in C*+‘, X(t) A X’(t). The osculating space of order p to V along the generator X(t) A X’(t) is

x(t) A x’(t) A . . . A Xcp+l)(t).

Hence we have a homomoprhism of vector bundles over fCp), Ap + 1 f! E, ,N + Ep + z .N _ ~ + ,. p < r, and by the Riemann-Hurwitz formula

dP = K[f’P”C1(Ep+2,N-p+2) - ~I(A~+I.~!EI,N)I

= -vp+(p+2)vo- (P + HP + 2) X(&l )

2 1 7 q.e.d.

Another kind of developable surface is the cone.


PROPOSITION 4.7. Let f: M, --t PN be an analytic curve in a complex projectitie space

which is not everywhere singular of order r and let y E P, be a point not on Ml. Then if’ V is

the cone formed by joining y to all the points of Ml by lines, vO the order off, vp the order qf ftp) (the map which assigns to each point of M, the osculating space of order p along the

generator through that point) and if d, is the total singularity oj‘order p off, then

dp = -VP + (p + ljv, P(P + 1)

- - X(fM,). 2

Proof. If M is given locally by the N + l-vector function X(t), and if A is a vector in CN+’ such that 7(A) = y, where 7 is the map which assigns to each point of C”+’ other than the origin the line joining it to the origin, then the cone V is given by the family of planes in CN+‘A A X(t). The osculating space of orderp along A A X(t) is given by A A X(t) A . . . A Xtp)t. Therefore we have a vector bundle homomorphism over f (p)

(i8Apf’El,~)jEp+2,~-p--1

where 1 is the product line bundle, and, by the Riemann-Hurwitz formula

d P = ~[j-‘~‘*c~(E P+Z,N-P-1 ) - c,(I @ A,f !E,, r&l

= -v,+(p+ l)v,-- p(p + l) I 2 , q.e.cl.

Throughout this section we have given formulas which relate characteristic classes to the singularities of various orders of maps into projective spaces, which singularities are loci where the ranks of certain vector bundle homomorphisms fall. There is, however, another kind of singularity, for example double points, double tangents, double osculating spaces. We note the problem of finding formulas which relate these singularities to the other invariants.

5IV.3. Submanifolds of Tori.

Let V, be a complex torus; i.e., V, = C”/r where I is a maximal discrete subgroup of n-dimensional complex affine space. Recall $11, where we have defined a vector-bundle homomorphism urp : T,(C”) -+ T,(C”), where Tp is the bundle of osculating spaces of order p. Now UJ~ commutes with an affine transformation of C”, and therefore gives rise to a

map mp : T,< V,) + Tl(V,). But the tangent bundle of V, is a product bundle, since V, is a Lie group; let S, be the tangent space at the origin and let 5 : T,( V,,) + S, be the translation of tangent vectors to the origin.

Now let f: A4 + V” be a complex-analytic map. Then f induces f, : T,(M) + Tp( V,), the p-th differential of J

DEFINITION. m,f,(T,(M)I,) we caN the osculating space of order p to f: iU -+ V, at x.

Clearly the local theory of the osculating spaces of submanifolds of a torus coincides with that for submanifolds of affine spaces, and is a special case of that for submanifolds of projective spaces. Thus we can say that f: M --) V, is non-singular of order p at x if mpfp : T,(M)/, + T,(V,) is injective.


DEFINITION. Let f be non-sing&r of order p. Then we call the map f (p’ : M,,,

+ Gv~rn p) n -v(m.p)y P I defined b S’p’W = ~~,f,U’,(~,,,)~,), where Gvcn,,pj.n -vlm,pj is the Grass- mann manifold of v(m, p)-planes through the origin of S,, the p-th associated map to J

THEOREM 4.8. (The Plficker Formulas for CurLIes in a Torus). Let f: M, + V, be a

complex-analytic map of a complex-analytic manifold of dimension 1 into a complex torus,

which is not totally singular of order r. Then if wp is the stationary index of rank p, and vp the

order off(P),

- wo + VI = x(M,)

-WI + v2 - 2v, = x(M 1)

-wp+vp+,-2v,+v,_,=~(M,),1<p~r.

Proof. We have a vector bundle homomorphism over f (p),

f(P)

T,@fJ -E - P.” P

Ml-G _ P.” P

where f (p)’ is defined by f (p)‘(x’) = ( f(P)n(~‘), <mpfp(x')). Let d, denote the total singularity of order p off. By the Riemann-Hurwitz formula

d, = -vp - K[~,(T,(M))] = -17~ - ~&WI).

The result now follows from the computation of Theorem 4.4 with v. set equal to zero. Maps such as are treated in this theorem exist. For let M be a compact Riemann

surface of genus p > 1. Then there is a canonical map of M into its Jacobian variety, which is a complex torus of dimension p, given by the abelian integrals. The singular points of order p of this map may be seen to be precisely the Weierstrass points of M.

1. 2.

3.

4. 5. 6.

7. .

8. 9.

REFERENCES

W. AMBROSE, R. S. PALAIS and I. M. SINGER: Sprays, Ann. Acad. Bras. Sci. 32 (1960), 163-178. M. F. ATIYAH: Complex analytic connections in fibre bundles, Trans. Amer. Math. Sot., 85 (1957). 181-207. E. CARTAN: Sur les variktts de courbure constante d’un espacz euclidien ou non-euclidien, Bull. Sot. Math. Fr. 47 (1919). 125-160: 48 (1920). 132-208: Oeuures ComultVes, Partie III, 1, 321-432. H. CARTAN and S. EILENBERG’: Homol&ical Algebra, Princeton University Press, 1956. S. S. CHERN: Differentiable Manifolds, Mimeographed notes, University of Chicago, 1953. S. S. CHERN: Geometrical structures on manifolds, Colloquium Lectures, American Mathematical Society, 1960. S. S. CHERN: Geometry of submanifolds in a complex projective space, Symposium International de Topologia Algebraica, pp. 87-96 Mexico City, 1958. G. DE RHAM: Vari&s Diflrentiables, Hermann, Paris, 1955. C. EHRESMANN: Les prolongements d’une variktt diffkentiable-I: Calcul des jets, prolongement principal; II: L’espace des jets d’order r de V, dans V,, C. R. Acad. Sci., Paris 233 (1951), 598400: 777-779.

DIFFERENTIAL GEOMETRY OF HJGHER ORDER 211

IO. B. L. FOSTER: Second order tangent vectors and derivations, Not. Amer. Math. Sot. 7 (1960), 949-950. 11. A. GROTHENDIECK: La thtorie des classes de Chem, Bull. Sot. Math. Fr. 86 (1958), 137-154. 12. M. W. HIRSCH: Immersions of manifolds, Trans. Amer. Math. Sot. 93 (1959), 242-276. 13. F. HIRZEBRUCH : Neue Topologische Methoden in der Atgebraischen Geometrie, Springer, Berlin, 1956. 14. H. I. LEVINE: Singularities of’ dtferential mappings, duplicated notes, Mathematisches Institut, Bonn

University. 15. S. NAKANO: Tangential vector bundle and Todd canonical systems of an algebraic variety, Mem. Coil.

Sci. Kyoto A, 29, 2 (1955), 145-149. 16. J. P. SERRE: Faisceaux algebriques cohtrents, Ann. Math., Princeton 61 (1955), 197-278. 17. S. STERNBERG and R. G. SWAN: On maps with non-negative Jacobian, Mich. Math. J. 6 (1959), 339-342. 18. E. THOMAS: On tensor products of n-plane bundles, Arch. Math.. Karlsruhe 10 (1959). 174-179. 19. R. THOM: Les ensembles singuliers dune application differentiable et leurs proprittes homologiques,

Seminaire de Topologie de Strasbourg, 1957. 20. A. WEIL: Theorie des points proches sur les varietes differentiables, GPome’tri~ Diferentielle, Colloques

Zvternationaux du C. N. R. S., pp. 11 l-l 17, Strasbourg, (1953). 21. H. WEYL and J. WEYL: Meromorphic Functions and Analytic Curves, Ann. Math. Stud. No. 12,

Princeton University Press, 1943. 22. H. WHITNEY: On regular closed curves in the plane Compos. Math. 4 (1937), 276-284.

Massachusetts Instifute of Technology, Cambridge, Massachussets, U.S.A.

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